The need to make a priori decisions about the level of approximation that can be accepted — and subsequently justified — in flows of industrial complexity is a perennial problem for CFD analysts. This problem is particularly acute in the simulation of rotating cavity flows, where the stiffness of the equation set results in protracted convergence times, making any simplification extremely attractive. For example, it is common practice, in applications where the geometry and boundary conditions are axisymmetric, to assume that the flow solution will also be axisymmetric. It is known, however, that inappropriate imposition of this assumption can lead to significant errors. Similarly, where the geometry or boundary conditions exhibit cyclic symmetry, it is quite common for analysts to constrain the solutions to satisfy this symmetry through boundary condition definition. Examples of inappropriate use of these approximating assumptions are frequently encountered in rotating machinery applications — such as the ventilation of rotating cavities within aero-engines. Objective criteria are required to provide guidance regarding the level of approximation that is appropriate in such applications. In the present work, a study has been carried out into: • The extent to which local 3-D features influence solutions in a generally 2-D problem. Criteria are proposed to aid in decisions about when a 2-D axisymmetric model is likely to deliver an acceptable solution. • The influence of flow features which may have a cyclic symmetry that differs from the bounding geometry or imposed boundary conditions (or indeed have no cyclic symmetry). • The influence of unsteady flow features and the extent to which their effects can be represented by mixing plane or multiple reference frame approximations.
A Priori Convergence Estimates for a Rough Poisson-Dirichlet Problem with Natural Vertical Boundary Conditions
